Two-Dimensional Localization Detected by the Imaginary Vector Potential

Abstract

In non-interacting disordered systems with the timereversal symmetry, all the states are believed to be localized in two or lower dimension. The role of the nonHermitian effect such as the relaxation processes breaks the phase coherence, and provides a gradual crossover into metallic behavior. An exception to the rule is the non-Hermitian term coupled with the first-order differential, i.e., the imaginary vector potential. It has been recognized subsequently that sufficiently large imaginary vector potential always induces a delocalization transition (not crossover) with complex eigen energies. The purpose of the paper is to establish a new way of probing the localization property by using this nonHermitian effect as a probe. Unlike previous attempts to estimate the localization length from the complex energy spectrum, our procedure is on principle applicable to any dimension. We confirm the validity by examining the 2D systems numerically. It is shown that the breakdown of the imaginary gauge transformation can characterize the inverse localization length near the band center, though we need to clarify its meaning. The non-Hermitian delocalization process is far from complete understanding in the twoand higher-dimensions, e.g., whether the transition is continuous or abrupt. On this regard, we also briefly mention our view based on our numerical results. The Hamiltonian considered for numerical calculations is the 2D non-Hermitian tight-binding model,